Biderivations of the higher rank Witt algebra without anti-symmetric condition

Let F be a eld of characteristic zero. We denote by Z the sets of all integers. We x a positive integer d ≥ 1 and denote by Wd the derivation Lie algebra of the Laurent polynomial algebra A = F [z±1 1 ,⋯, z±1 d ] in d commuting variables z1,⋯, zd over F. It is well known that the in nite-dimensional Lie algebraWd is called the Witt algebra of rank d. Its representations have attracted a lot of attention from many mathematicians [1–5]. According to their notations, the Witt algebra can be described as follows. For i ∈ {1, 2, ..., d}, set ∂i = zi ∂ zi . For any n ∈ Z d (considered as row vectors, i.e., n = (n1,⋯, nd) where ni ∈ Z), set zn = zn1 1 z n2 2 ⋯z nd d . We x the vector space F d of 1 × d matrices. Denote the standard basis by e1, e2, ..., ed, which are the row vectors of the identity matrix Id. Let (⋅, ⋅) be the standard symmetric bilinear form such that (u, v) = uvT ∈ F, where vT is the matrix transpose. For u ∈ Fd and r ∈ Zd, we denote D(u, r) = z∑i=1 ui∂i. Then we have [D (u, r) , D (v, s)] = D (w, r + s) , u, v ∈ F , r, s ∈ Z , (1)


Introduction
Let F be a eld of characteristic zero. We denote by Z the sets of all integers. We x a positive integer d ≥ and denote by W d the derivation Lie algebra of the Laurent polynomial algebra A = F z ± , ⋯, z ± d in d commuting variables z , ⋯, z d over F. It is well known that the in nite-dimensional Lie algebra W d is called the Witt algebra of rank d. Its representations have attracted a lot of attention from many mathematicians [1][2][3][4][5]. According to their notations, the Witt algebra can be described as follows.
For i ∈ { , , ..., d}, set ∂ i = z i ∂ z i . For any n ∈ Z d (considered as row vectors, i.e., n = (n , ⋯, n d ) where n i ∈ Z), set z n = z n z n ⋯z n d d . We x the vector space F d of × d matrices. Denote the standard basis by e , e , ..., e d , which are the row vectors of the identity matrix I d . Let (⋅, ⋅) be the standard symmetric bilinear form such that (u, v) = uv T ∈ F, where v T is the matrix transpose. For u ∈ F d and r ∈ Z d , we denote D(u, r) = z r ∑ d i= u i ∂ i . Then we have where w = (u, s) v − (v, r) u. Therefore, the Witt algebra of rank d is the F-linear space Recall that It is well-known that derivations and generalized derivations are very important subjects in the research of both algebras and their generalizations. In recent years, biderivations have interested a great number of authors, see [6][7][8][9][10][11][12][13][14][15][16]. In [7], Brešar et al. showed that all biderivations on commutative prime rings are inner biderivations, and determined the biderivations of semiprime rings. The notion of biderivations of Lie algebras was introduced in [15]. Since then, biderivations of Lie algebras have been studied by many authors. It may be useful and interesting for computing the biderivations of some important Lie algebras. In particular, the authors in [11] determined anti-symmetric biderivations for all W d . All biderivations of W without antisymmetric condition were later obtained in [14]. In the present paper, we shall use the methods of [14] to determine all biderivations of W d for all d ≥ .
Next, let us introduce the de nition of biderivation. For an arbitrary Lie algebra L, a bilinear map f ∶ L × L → L is called a biderivation of L if it is a derivation with respect to both components. Namely, for each x ∈ L, both linear maps φ x and ψ x form L into itself given by φ for all x, y, z ∈ L. Denote by B(L) the set of all biderivations of L. For λ ∈ C, it is easy to verify that the bilinear In this paper, we will prove that every biderivation of W d without anti-symmetric condition is inner. As an application, we characterize the commutative post-Lie algebra structures on W d .

Biderivations of the Witt algebras
We rst give some lemmas which will be useful for our proof.

Lemma 2.1 ([17]). Every derivation of W d is inner.
Then there are linear maps φ and ψ from W d into itself such that The proof is completed.
Proof. Lemma 2.2 tells us that for all i, j ∈ { , ⋯, d} and r, s ∈ Z d . From (3) and (4), the conclusion follows by direct computations.
We will prove that a It is clear that the right-hand side of (7) does not contain any non-zero elements in h, thereby the left-hand side is so. From this, one has that Thanks to s j ≠ , we have by (8)  Proof. We will only prove (10), the proof for (9) is similar. Continuing the use of the assumptions (3) and (4), we also have that (7) Therefore, from (11)  Proof. We use the assumptions (3) and (4). With Lemmas 2.4 and 2.5, Equation (5) becomes Although r ≠ , but we still can nd a subset {s ( ) , ⋯,s (d) } of Z d such thats ( ) , ⋯,s (d) are F-linearly independent withs (t) ≠ r, t = , ⋯, d. Let s run over the vectorss ( ) , ⋯,s (d) in (12), then we see that Similarly, we have b (j,s) , Next, by taking s = ( , ⋯, ) ≐ e in (13), we have a j,e for all i ≠ j. This tells us that a ,e for any i ≠ and a for all i, j ∈ Z. Namely, it follows that, in (3) and (4), a Note that Lemma 2.6 tells us that, in (3) and (4), a (i,r) k,n = δ i,k δ n,r λ and b (j,s) k,n = δ j,k δ n,s λ for any i, j ∈ Z and r, s ∈ Z d ∖ { }. All these together with letting r = in (5), deduce that = for every k ≠ i. This proves that φ(D(e i , )) = λD(e i , ). Similarly, we can obtain that ψ(D(e j , )) = λD(e j , ). The proof is completed.
Our main result is the following.

Theorem 2.8. Every biderivation of W d without anti-symmetric condition is inner.
Proof. Suppose that f is a biderivation of W d . Let φ be determined by Lemma 2.2, λ ∈ F be given by Lemma 2.7. Note that W d is spanned by D(u, ), D(u, r)  for all x, y ∈ W d , as desired.

An application
The anti-symmetric biderivation can be applied to linear commuting maps, commuting automorphisms and derivations, see [8]. Another application of biderivation without the anti-symmetric condition is the characterization of post-Lie algebra structures. Post-Lie algebras have been introduced by Valette in connection with the homology of partition posets and the study of Koszul operads [18]. As [19] point out, post-Lie algebras are natural common generalization of pre-Lie algebras and LR-algebras in the geometric context of nil-a ne actions of Lie groups. Recently, many authors have studied some post-Lie algebras and post-Lie algebra structures [19][20][21][22][23]. In particular, the authors of [19] study the commutative post-Lie algebra structure on Lie algebra. Let us recall the following de nition of a commutative post-Lie algebra.
De nition 3.1. Let (L, [, ]) be a Lie algebra over F. A commutative post-Lie algebra structure on L is a F-bilinear product x ○ y on L and satis es the following identities: x ○ y = y ○ x, for all x, y, z ∈ L. It is also said that (L, [, ], ○) is a commutative post-Lie algebra. Lemma 3.2 ([14]). Let (L, [, ], ○) be a commutative post-Lie algebra. If we de ne a bilinear map f ∶ L × L → L given by f (x, y) = x ○ y for all x, y ∈ L, then f is a biderivation of L. Theorem 3.3. Any commutative post-Lie algebra structure on the generalized Witt algebra W d is trivial. Namely, x ○ y = for all x, y ∈ W d .
Proof. Suppose that (W d , [, ], ○) is a commutative post-Lie algebra. By Lemma 3.2 and Theorem 2.8, we know that there is λ ∈ F such that x ○ y = λ[x, y] for all x, y ∈ W d . Since the post-Lie algebra is commutative, so we have λ[x, y] = λ[y, x]. It implies that λ = . The proof is completed.